The most useful rules of basic algebra,
free, simple, & intuitively organized

Howdy! Here are a few very handy rules of algebra. These basic rules are useful for everything from figuring out your gas mileage to acing your next math test — or even solving equations from the far reaches of theoretical physics. Happy calculating!

Algebra Rule 14

The result of a negative exponent is the inverse of the same positive exponent

```a^{-n} = {1\over a^n}```

It might seem odd to have a negative exponent (since you can't multiply something by itself a negative number of times). However, if we take a closer look at the rule ``a^na^m = a^{n+m}`` we can see that it implies that ``a^{-n}`` must equal ``{1 \over a^n}``, the multiplicative inverse or reciprocal of ``a^n``.

This becomes clear looking at the ``a^{n+m}`` side of the equation from rule 11. What happens if ``m`` is negative? Obviously, this will reduce the combined value of the exponent (for example, ``2^{4-2} = 2^2``). What does this mean for the left hand side of the ``a^na^m = a^{n+m}`` equation? It means that the value of, for example, ``2^4`` must be reduced to ``2^2`` when it is multiplied by ``2^{-2}``. If, as this rule states, ``a^{-n} = {1 \over a^n}``, this works out perfectly: ``2^4 * 2^{-2} = 2^4 * {1 \over 2^2} = 16 * {1 \over 4} = 4 = 2^2 = 2^{4-2}``

```2^{-2} = {1\over2^2} = {1\over 4}```
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A little bit about

Algebra rules is a project by two of the folks who run The Autodidacts.

A couple of autodidact math enthusiasts, we were looking for all the rules of basic algebra concisely presented in one place. We couldn’t find such a place, so we made

These simple rules — applied with a pinch of imagination and a dash of arithmetic — can divide, conquer, and solve just about any practical algebra problem.

If you find errata in the math, bugs in the code of, or just want to say Eh, please send us a letter or join us on our roost: @rulesofalgebra.


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